The Signature Kernel Is the Solution of a Goursat PDE

IF 1.9 Q1 MATHEMATICS, APPLIED SIAM journal on mathematics of data science Pub Date : 2020-06-26 DOI:10.1137/20M1366794
C. Salvi, Thomas Cass, J. Foster, Terry Lyons, Weixin Yang
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引用次数: 32

Abstract

Recently, there has been an increased interest in the development of kernel methods for learning with sequential data. The signature kernel is a learning tool with potential to handle irregularly sampled, multivariate time series. In"Kernels for sequentially ordered data"the authors introduced a kernel trick for the truncated version of this kernel avoiding the exponential complexity that would have been involved in a direct computation. Here we show that for continuously differentiable paths, the signature kernel solves a hyperbolic PDE and recognize the connection with a well known class of differential equations known in the literature as Goursat problems. This Goursat PDE only depends on the increments of the input sequences, does not require the explicit computation of signatures and can be solved efficiently using state-of-the-arthyperbolic PDE numerical solvers, giving a kernel trick for the untruncated signature kernel, with the same raw complexity as the method from"Kernels for sequentially ordered data", but with the advantage that the PDE numerical scheme is well suited for GPU parallelization, which effectively reduces the complexity by a full order of magnitude in the length of the input sequences. In addition, we extend the previous analysis to the space of geometric rough paths and establish, using classical results from rough path theory, that the rough version of the signature kernel solves a rough integral equation analogous to the aforementioned Goursat PDE. Finally, we empirically demonstrate the effectiveness of our PDE kernel as a machine learning tool in various machine learning applications dealing with sequential data. We release the library sigkernel publicly available at https://github.com/crispitagorico/sigkernel.
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签名核是一个Goursat PDE的解
最近,人们对开发用于序列数据学习的核方法越来越感兴趣。签名核是一种学习工具,具有处理不规则采样、多变量时间序列的潜力。在“顺序排序数据的内核”一文中,作者介绍了这个内核的截断版本的一个内核技巧,避免了直接计算中可能涉及的指数级复杂性。在这里,我们证明了对于连续可微路径,签名核解决了一个双曲PDE,并识别了与文献中称为Goursat问题的一类众所周知的微分方程的联系。此Goursat PDE仅依赖于输入序列的增量,不需要显式计算签名,并且可以使用状态- arthybolic PDE数值解算器有效地求解,为未截断的签名内核提供了一个内核技巧,具有与“序列有序数据的内核”方法相同的原始复杂性,但具有PDE数值方案非常适合GPU并行化的优点。它有效地将输入序列的复杂度降低了整整一个数量级。此外,我们将之前的分析扩展到几何粗糙路径空间,并利用粗糙路径理论的经典结果,建立了签名核的粗糙版本求解类似于上述Goursat PDE的粗糙积分方程。最后,我们通过经验证明了PDE内核在处理顺序数据的各种机器学习应用程序中作为机器学习工具的有效性。我们在https://github.com/crispitagorico/sigkernel公开发布了sigkernel库。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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